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## Tuesday, 26 March 2013

### Complex numbers as Matrices

In this section, we use matrices to give a representation of complex numbers. Indeed, consider the set

We will write

Clearly, the set  is not empty. For example, we have

In particular, we have

for any real numbers abc, and d.

Algebraic Properties of
1.
Addition: For any real numbers abc, and d, we have

Ma,b + Mc,d = Ma+c,b+d.

In other words, if we add two elements of the set , we still get a matrix in . In particular, we have

-Ma,b = M-a,-b.

2.
Multiplication by a number: We have

So a multiplication of an element of  and a number gives a matrix in .
2.
Multiplication: For any real numbers abc, and d, we have

In other words, we have

This is an extraordinary formula. It is quite conceivable given the difficult form of the matrix multiplication that, a priori, the product of two elements of  may not be in  again. But, in this case, it turns out to be true.

The above properties infer to  a very nice structure. The next natural question to ask, in this case, is whether a nonzero element of  is invertible. Indeed, for any real numbers a and b, we have

So, if , the matrix Ma,b is invertible and

In other words, any nonzero element Ma,b of  is invertible and its inverse is still in  since

In order to define the division in , we will use the inverse. Indeed, recall that

So for the set , we have
Ma, b÷Mc, d = Ma, b×Mc, d-1 = Ma, b×M,

which implies
Ma, b÷Mc, d = M, -

The matrix Ma,-b is called the conjugate of Ma,b. Note that the conjugate of the conjugate of Ma,b is Ma,b itself.

Fundamental Equation. For any Ma,b in , we have

Ma,b = a M1,0 + b M0,1 = a I2 + b M0,1.

Note that

M0,1 M0,1 = M-1,0 = - I2.

Remark. If we introduce an imaginary number i such that i2 = -1, then the matrix Ma,b may be rewritten by

a + bi